abstract let s c and l - adventist university of the...
TRANSCRIPT
July 2016 | Vol. 19 No. 2
AUP Research Journal | ISSN 1655-5619 80
Theorems on nth Dimensional Laplace TransformMichael Sta. Brigida1, Edwin Balila2, Edna Mercado3
1Cavite State University2Adventist University of the Philippines
3De La Salle University - Dasmariñas
Abstract
Let U be the set of all functions from [0,∞)n to and V be the set of all functions from S C n to . Then the nth dimentional Laplace transform is the mapping
Ln : U → Vdefined by:
Where In this paper we gave alternative proof for some theorems on properties of nth dimensional Laplace Transform, we proved that if is piecewise continuous on [0,∞)n and function of exponential order then the nth dimensional Laplace transform defined above exists, absolutely and uniformly convergent, analytic and infinitely differentiable on Re (s1) > 1, Re (s2) > 2, ..., Re (s2) > n, and we gave also some corollaries of these results.
Keywords: theorems, Laplace transform
Laplace transform is a widely used integral trans-form with many applications in Physics, Engineer-ing and Applied Mathematics. It is named after Pierre-Simon Laplace, who introduced the trans-form in his work on probability theory. It is also very useful for solving ordinary and partial differ-ential equations, difference equations, integral equations as well as mixed equations. In physics and engineering, it is used for analysis of linear time-invariant systems such as electrical circuits, harmonic oscillators, optical devices, and me-chanical sytems. It can be used as a very effective tool in simplifying the calculations in many fields of engineering and mathematics. It also provides a powerful method for analyzing linear systems.
Estrin and Higgins [21] in 1951 gave the systematic account of the general theory of an operational calculus using double Laplace trans-form and illustrated its application through solu-
tion of two problems, one in electrostatics, the other in heat conduction. Coon and Bernstein [22] conducted systematic study of double La-place transform in 1953 and developed its prop-erties including conditions for transforming de-rivatives, integrals, and convolution. Buschman [23] in 1983 used double Laplace transform to solve a problem on heat transfer between a plate and a fluid flowing across the plate. In 1989, Debnath and Dahiya [11], [12] established a set of new theorems concerning multidimensional Laplace transform of some functions and used the technique developed to solve an electrostat-ic potential problem. In 1992, Brychov, Glaeske, Prudnikov, and Tuan [27] in their book entitled Multidimensional Integral Transformations dis-cussed and proved several theorems on the properties of nth dimensional Laplace transform. In 2013, Aghili and Zeinali [24] implemented mul-
Vol. 19 No. 2 | July 2016
AUP Research Journal | ISSN 1655-561981
tidimensional Laplace transform method for solv-ing certain non-homogeneous forth order partial differential equations. Atangana [26] discussed some properties of triple Laplace transform and applied to some kind of third order partial dier-ential equations. In 2014, Eltayeb, Kilicman, and Mesloub [18] applied double Laplace transform method to evaluate the exact value of double innite series. Hamood Ur Rehman, Muzammal Iftikhar, Shoaib Saleem, Muhammad Younis, and Abdul Mueed [19] discussed some properties and theorems about the quadruple Laplace transform and gave a good strategy for solving the fourth order partial differential equations in engineer-ing and physics by quadruple Laplace transform. Dhunde and Waghmare [20] presented the ab-solute convergence and uniform convergence of double Laplace transform and solved a Volterra Integro Partial Differential Equation using double Laplace transform.
Research ObjectiveThe main objective of this paper was to
extend some results in [10], [19], [20] [24] and [26], provide alternative proof for some results in [27] on theorems on the properties of the nth dimensional Laplace transform and derive some special multiple improper integral identities.
THE nth DIMENSIONAL LAPLACE TRANS-FORM
In this section we will deal with the main results. For brevity, we will use the following no-tation throughout this chapter.
Definiftion 1. Let U be the set of all func-tions from [0, )n to and V be the set of all functions from S to . Then the nth dimen-sional Laplace transform is the mapping
defined by:
Where
From the above definition, it is clear that
Occasionally, we also denote
It is also clear that there are some func-tions in U in which the nth dimensional Laplace transform does not exist. We now define a class of functions needed for the existence of the nth dimensional Laplace transform.
Definition 2. A function is called piecewise continous in the region if the region can be divided into a finite number of sub-region such that is continuous.
Definition 2 is a natural extension to nth
dimension of piecewise continuity and from this definition we can evaluate the nth dimentional La-place Transform using theorems on multiple in-tergrals. We now define other class of functions needed for the existence of nth dimensional La-place transform.
Definition 3. If positive real constant M and a vector in exist such that at least one i of xi > Ni is true for i = 1, 2, 3..., n we have:
then is said to be function of exponential order Definition 3 is an extension to nth dimen-sion for the functions of exponential order. Func-tions of exponential order cannot grow in abso-lute value more rapidly than if at least one of increases. Clearly, the exponential function
has exponential order =
July 2016 | Vol. 19 No. 2
AUP Research Journal | ISSN 1655-5619 82
(a1, a2, ..., an), whereas has exponen-tial order = where at least one of > 0 and any Bounded functions like sin , cos
, tan-1 have exponential order = (0, 0, ..., 0) where and
are continuous functions from to while has order = (-1, -1, ..., -1).
However, does not have expo-nential order.
CONVERGENCE THEOREM The following theorem gives us sufficient conditions for the existence of nth dimensional La-place transform. Theorem 1. If is piecewise contin-uous in every bounded region and of exponential order then the nth dimensional Laplace trans-form exists for all Re(s1) > , Re(s2) > ,...., Re(sn) > . Proof: From Definition 1 we have =
Hence we get:
where is a bound-ed region, is an unbounded region such that
and Clearly ,the integral
exists, since is piecewise continuous in ev-ery bounded region.So, we only need to show that the integral
exists.Now, we have a chain of inequality:
but since is of exponential order
where now the integral:
is equal to
Hence, if
then:
Therefore, if exists.
We now prove the absolute convergence of the nth dimensional Laplace integral.
Theorem 2. If is piecewise contin-uous on [0, )n and of exponential order then the nth dimensional Laplace transform:
is absolutely convergent for
Proof:From Theorem 1
if
thus
83
Vol. 19 No. 2 | July 2016
Theorems on nth Dimensional Laplace Transform
is absolutely convergent for all
Similar result for Theorem 2 can be found in [22] where the function is considered a map-ping from [0, )n to while in this result we con-sidered our function to be a mapping [0, )n to with additional condition that the function
must be piecewise continuous on [0, )n. In this result, we actually found the bound for the integral
We now prove two lemmas needed for the proof of the uniform convergence of the nth dimension-al Laplace transform.
Lemma 1. If is piecewise continu-ous on [0, )n and of exponential order such that , then
is piecewise continuous on [0, )n 1and of exponential order , Where are n - 1 dimensional vectors without and .
Proof:The function
is piecewise continuous on [0, )n, since is piecewise continuous on [0, )n, so it follows that is also piecewise continuous on [0, )n 1. we need only to show that is a function of exponential order , now
Hence
But it is clear that
is less than
Therefore,
is less than
for at least one of 1, 2, 3, ..., nthus,
That is, is of exponential order
Lemma 2. If is piecewise continuous on [0, )n and of exponential order then
is uniformly convergent on for
Proof:We know that
and from Lemma 1 we have:
provided for 1, 2, ..., n.whenever
then
July 2016 | Vol. 19 No. 2
AUP Research Journal | ISSN 1655-5619 84
So, we choose
,
this N is always positive for that is given any , there exists a value N > 0 such that
whenever for all values of with
This is precisely the condition required for the uniform convergence in the region
The following theorem proves the uni-form convergence of the nth dimensional Laplace transform.
Theorem 3. If is piecewise contin-uous on [0, )n and of exponential order then
converges uniformly on
Proof:Without loss of generality, write as
where
now by Lemma 2,
converges uniformly on Hence is piecewise continuous, by Lemma 1, on [0, )n-1 and of exponential order
so again by Lemma 2 the integral
converges uniformly on
Thus, by applying Lemma 1 and Lemma 2 repeat-edly we see that the intergral
is uniformly convergent on
Similar result for Theorem 3 can be found in [27]. In our result, we used two lemmas to prove the uniform convergence of the nth dimentional La-place integral.
In what follows, we will state that based on Theorem 3, the order limit and integral can safely be reversed. The proof is straightforward.
COROLLARY 1. If is piecewise con-tinuous on [0, )n and exponential order then
is continuous and integrable complex valued func-tion in real variables on
Other implication of Theorem 3 is again stated as corollary.
COROLLARY 2. If the integral is convergent, then
converges uniformly on for
BEHAVIOR OF THE nth DIMENSIONAL LAPLACE TRANSFORM
From Corollary 1, we can insert the limit on any point on the region
inside the nth dimensional Laplace integral and thus
can be expressed as
85
Vol. 19 No. 2 | July 2016
Theorems on nth Dimensional Laplace Transform
which can be further simplified as
ANALYTICITY OF THE nth DIMENSIONALLAPLACE TRANSFORM Analyticity plays an important role in
complex analysis and in physics, since once the function was determined to be analytic in some region then its higher order derivatives are also analytic in the same region. Also the real and imaginary parts of the function are definitely many times differentiable and harmonic in the same region.
Before proving the analyticity of nth di-mensional Laplace transform we first prove the following lemmas.
LEMMA 3. If is piecewise continu-ous on [0, )n and of exponential order then the integrals
converges uniformly on , where
Proof of (a):By direct evaluation we get
for But the integral
converges for Thus, by Weiertrass M test, the integral
converges uniformly on
Proof of (b): The proof is analogous to the proof of (a).
We now prove the analyticity of nth dimensional Laplace transform
Theorem 4. If is piecewise contin-uous on [0, )n and of exponential order then
is analytic function on
Proof: Let , then
where and are re-spectively expressed as
and
By Lemma 3, we can reverse the order of derivative and the integrals of the real and imagi-nary parts of and hence the integrals
where
and
and uniformly convergent on
Thus,
are continuous on for
July 2016 | Vol. 19 No. 2
AUP Research Journal | ISSN 1655-5619 86
and also, we see that:
and
that is, Cauchy-Riemann equations are satisfied on for Thus,
is analytic function on
Similar result for Theorem 4 can be found in [27]. In our result, we used Lemma 3 to prove the analyticity of the nth dimensional Laplace transform. From Theorem 4 it follows that the nth dimensional Laplace transform is infinitely many times differentiable and continuous on
and hence can be differentiated of all orders in-side the nth dimensional Laplace integral operator.
REFERENCES[1] R. Wrede and M. R. Spiegel. Schaum’s Out-
lines Advanced Calculus, 2010.
[2] M. R. Spiegel, S. Lipschutz, J. J. Schiller, and D. Spellman. Schaum’s Outlines Complex Variables, 2009.
[3] J. L. Schiff. The Laplace Transform Theory and Applications, 1999.
[4] M. R. Spiegel. Schaum’s Outlines Laplace Trans-forms, 1965.
[5] W. F. Trench. Functions defined by improper integrals. In Introduction to Real Analysis, 2013.
[6] R. Merris. Combinatorics, 2003.
[7] W. F. Trench. Introduction to Real Analysis, 2013.
[8] S. Lipschutz and M. L. Lipson. Schaum’s Out-
lines Theory and Problems of Linear Al-gebra, 2013.
[9] A. Kilicman and H. E. Gadain. On the applica-tions of Laplace and Sumudu transforms.Journal of the Franklin Institute. (347), 5:848-862, 2010.
[10] H. Eltayeb and A. Kilicman. A note on dou-ble Laplace transform and telegraphic equations. Hindawi Publishing Corpora-tion Abstract and Applied Analysis. Article ID 932578, 6 pages, 2013.
[11] A. Babakhani and R. S. Dahiya. Systems of multidimensional Laplace transforms and a heat equation. In Proceedings of the 16th Conference on Applied Mathematics, vol. 7 of Electronic Journal of Differential Equations, pp. 2536, University of Central Oklahoma, Edmond, Oklahoma, USA, 2001.
[12] R. S. Dahiya and J. S. Nadja. Theorems on n- dimensional Laplace transforms and their applications. 15th Annual Conference of Applied Mathematics, Univ. of Central Oklahoma. Electronic Journal of Differen-tial Equations, 02:61-74. 1999.
[13] H. Eltayeb, A. Kilicman, and P. R. Agarwal. An analysis on classifications of hyperbolic and elliptic PDEs. Mathematical Sciences, 6:47, 2012.
[14] V. G. Gupta and B. Sharma. Application of Sumudu transform in reaction-diffusion systems and nonlinear waves. Applied Mathematical Sciences, (4), 9:435-446, 2010.
[15] A. Kilicman and H. Eltayeb. On the applica-tions of Laplace and Sumudu transform. Journal of the Frankin Institute, (347), 5:848-862, 2010.
87
Vol. 19 No. 2 | July 2016
Theorems on nth Dimensional Laplace Transform
[16] A. Kilicman, H. Eltayeb, and P. R. Agarwal. On Sumudu transform and system of di-erential equations. Abstract and Applied Analysis, Article ID 598702, 11 pages, 2010.
[17] A. Kilicman and H. E. Gadain. An application of double Laplace transform and double Sumudu transform. Lobachevskii Journal of Mathematics, (30), 3:214-223, 2009.
[18] H. Eltayeb, A. Kilicman, and S. Mesloub. Ex-act evaluation of innite series using dou-ble Laplace transform technique. Abstract and Applied Analysis, 2014. http://dx.doi.org/10.1155/2014/327429 (2014).
[19] H. U. Rehman, M. Iftikhar, S. Saleem, M. Younis, and A. Mueed. A computation-al quadruple Laplace transform for the solution of partial differential equations,.Applied Mathematics, 5:3372-3382. 2014. http://www.scirp.org/journal/am
h t t p : / / d x . d o i . o r g / 1 0 . 4 2 3 6 /am.2014.521314
[20] R. R. Dhunde and G. L. Waghmare. Some convergence theorems of double Laplace transform. Journal of Informatics and Mathematical Sciences, (6), 1:45-54, 2014 .ISSN 0975-5748 (online); 0974-875X (print) RGN Publications.
[21] T. A. Estrin and T. J. Higgins. The solution of boundary value problems by multi-ple Laplace transformation. Journal of the Franklin Institute, (252), 2: 153-167, 1951.
[22] G. A. Coon and D. L. Bernstein. Some prop-erties of the double Laplace transforma-tion. American Mathematical Society, 74: 135-176, 1953.
[23] R. G. Buschman. Heat transfer between a flu-id and a plate: Multidimensional Laplace transformation methods. Interat. J. Math. and Math. Sci., (6):3, 589-596, 1983.
[24] A. Aghili and H. Zeinali. Two-dimensional Laplace transforms for non-homogeneous fourth-order partial differential equa-tions.Journal of Mathematical Research and Applications, (1), 2:48-54, December, 2013.
[25] M. M. Moghadam and H. Saeedi. Application of differential transforms for solving the Volterra integro-partial dierential equa-tions. Iranian Journal of Science and Tech-nology, Transaction A-34 (A1), 2010.
[26] A. Atangana. A note on the triple La-place transform and its applica-tions to some kind of third-order differential equation. Hindawi Pub-lishing Corporation Abstract and Applied Analysis, Article ID 769102, 10 pages, 2013. http://dx.doi.org/10.1155/2013/769102
[27] Y. A. Brychov, H. J. Glaeske, A. P Prudnikov, and V. K. Tuan. Multidimentional integral transformations. Gordon and Breach Sci-ence Publishers, 1992.